Element in Left Coset iff Product with Inverse in Subgroup

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Let $G$ be a group.

Let $H$ be a subgroup of $G$.

Let $x, y \in G$.

Let $y H$ denote the left coset of $H$ by $y$.


$x \in y H \iff x^{-1} y \in H$


\(\ds x\) \(\in\) \(\ds y H\)
\(\ds \leadstoandfrom \ \ \) \(\ds \exists h \in H: \, \) \(\ds x\) \(=\) \(\ds y h\) Definition of Left Coset
\(\ds \leadstoandfrom \ \ \) \(\ds \exists h \in H: \, \) \(\ds x^{-1}\) \(=\) \(\ds h^{-1} y^{-1}\) Inverse of Group Product
\(\ds \leadstoandfrom \ \ \) \(\ds \exists h \in H: \, \) \(\ds x^{-1} y\) \(=\) \(\ds h^{-1}\) Product with $y$ on the right
\(\ds \leadstoandfrom \ \ \) \(\ds x^{-1} y\) \(\in\) \(\ds H\) $H$ is a subgroup


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